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This paper studies the assortment optimization problem under the General Luce Model (GLM), a discrete choice introduced by Echenique and Saito (2015) that generalizes the standard multinomial logit model (MNL). The GLM does not satisfy the Independence of Irrelevant Alternatives (IIA) property but it ensures that each product has an intrinsic utility and uses a dominance relation between products. Given a proposed assortment $S$, consumers first discard all dominated products in $S$ before using a MNL model on the remaining products. The General Luce Model may violate the traditional regularity condition, which states that the probability of choosing a product cannot increase if the offer set is enlarged. As a result, the model can model behaviour that cannot be captured by any discrete choice model based on random utilities. The paper proves that the assortment problem under the GLM is polynomially-solvable. Moreover, it proves that the capacitated assortment optimization problem under the General Luce Model (GLM) is NP-hard and presents polynomial-time algorithms for the cases where (1) the dominance relation is utility-correlated and (2) its transitive reduction is a forest. The proofs exploit a strong connection between assortments under the GLM and independent sets in comparability graphs.
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Assortment and Price Optimization
Under the Two-Stage Luce model
Alvaro FloresGerardo BerbegliaPascal Van Hentenryck
Monday 22nd April, 2019
Abstract
This paper studies assortment and pricing optimization problems under the Two-Stage Luce
model (2SLM), a discrete choice model introduced by Echenique and Saito (2018) that gener-
alizes the multinomial logit model (MNL). The model employs an utility function as in the the
MNL, and a dominance relation between products. When consumers are offered an assortment
S, they first discard all dominated products in Sand then select one of the remaining products
using the standard MNL. This model may violate the regularity condition, which states that
the probability of choosing a product cannot increase if the offer set is enlarged. Therefore, the
2SLM falls outside the large family of discrete choice models based on random utility which con-
tains almost all choice models studied in revenue management. We prove that the assortment
problem under the 2SLM is polynomial-time solvable. Moreover, we show that the capacitated
assortment optimization problem is NP-hard and but it admits polynomial-time algorithms for
the relevant special cases cases where (1) the dominance relation is attractiveness-correlated
and (2) its transitive reduction is a forest. The proofs exploit a strong connection between
assortments under the 2SLM and independent sets in comparability graphs. Finally, we study
the associated joint pricing and assortment problem under this model. First, we show that well
known optimal pricing policy for the MNL can be arbitrarily bad. Our main result in this sec-
tion is the development of an efficient algorithm for this pricing problem. The resulting optimal
pricing strategy is simple to describe: it assigns the same price for all products, except for the
one with the highest attractiveness and as well as for the one with the lowest attractiveness.
1 Introduction
Revenue Management (RM) is the managerial practice of modifying the availability and the prices
of products in order to maximise revenue or profit. The origin of this discipline dates back to
the 1970’s, following the deregulation of the US airline market. A large volume of research has
been devoted to this area over the last 45 years, with successful results in many industries ranging
College of Engineering & Computer Science, Australian National University, Australia.
Melbourne Business School, The University of Melbourne, Australia.
H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology, USA.
1
from airlines, hospitality, retailing, and others (McGill and van Ryzin, 1999; K¨ok, Fisher, and
Vaidyanathan, 2005; Vulcano, van Ryzin, and Chaar, 2010).
Two main problems lay in the core of RM theory and practice: the optimal assortment problem,
and the pricing problem. The optimal assortment problem consists of selecting a subset of products
to offer customers in order to maximize revenue. Consider, for example, a retailer with limited space
allocated to mobile phones. If the store has more than 500 mobile phones that can be acquired
through its distributors (in various combinations of brands and sizes) and the mobile phone aisle
has capacity to fit 50 phones on the shelves, the store manager has to decide which subset of
products to offer given the product costs and the customer preferences.
In order to solve the assortment problem we need a model to predict how customers select
products when they are presented with a set of alternatives. Most models of discrete choice theory
postulate that consumers assign an utility to each alternative and given an offer set, they would
choose the alternative with maximum utility. Different assumptions on the distribution of the
utilities lead to different discrete choice models: Celebrated examples include the multinomial logit
(MNL) (Luce, 1959), the mixed multinomial logit (MMNL) (Daly and Zachary, 1978), and the
nested multinomial logit (NMNL) (Williams, 1977).
The multinomial logit model (MNL), also known as the Luce model, is widely used in discrete
choice theory. Since the model was introduced by Luce (1959), it was applied to a wide variety of
demand estimation problems arising in transportation (McFadden, 1978; Catalano, Lo Casto, and
Migliore, 2008), marketing (Guadagni and Little, 1983; Gensch, 1985; Rusmevichientong, Shen, and
Shmoys, 2010), and revenue management (Talluri and Van Ryzin, 2004; Rusmevichientong, Shen,
and Shmoys, 2010). One of the reasons for its success stems from its small number of parameters
(one for each product): This allows for simple estimation procedures that generally avoids over
fitting problems even when there is limited historical data (McFadden, 1974). However, one of the
flaws of the MNL is the property known as the Independence of Irrelevant Alternatives(IIA), which
states that the ratio between the probabilities of choosing elements xand yis constant regardless
of the offered subset. This property does not hold when products cannibalize each other or are
perfect substitutes (Ben-Akiva and Lerman, 1985; Debreu, 1960; Anderson, Depalma, and Thisse,
1992).
Several extensions to the MNL model have been introduced to overcome the IIA property and
some of its other weaknesses; They include the nested multinomial logit and the latent class MNL
model. These models however do not handle zero-probability choices well. Consider two products
aand b: The MNL model states that the probability of selecting aover bdepends on the relative
attractiveness of acompared to the attractiveness of b. Consider the case in which bis never
selected when ais offered. Under the MNL model, this means that bmust have zero attractiveness.
But this would prevent bfrom being selected even when ais not offered in an assortment.
On the other hand, the pricing problem amounts to determine the prices that a company should
offer, in order to best meet its objectives (profit maximization, revenue maximization, market share
maximization, etc.), while taking into consideration how customers will respond to different prices
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and the interaction between price and the intrinsic features that each product possess.
This paper considers both problems mentioned before, for the case when customers follow the
Two-Stage Luce model (2SLM). The 2SLM was recently introduced by Echenique and Saito (2018)
and unlike the MNL, it allows for violations to the IIA property and regularity (Berbeglia and
Joret, 2017). The Two-Stage Luce model generalizes the MNL by incorporating a dominance (anti-
symmetric and transitive) relation among the alternatives. Under such relationship, the presence
of an alternative xmay prevent another alternative yfrom being chosen despite the fact that both
are present in the offered assortment. In this case, alternative xis said to dominate alternative y.
However, when xis not present, ymight be chosen with positive probability if it is not dominated
by any other product z.
An important application of the 2SLM can be found in assortment problems where there exists
a direct way to compare the products over a set of features. For illustration, consider a telecommu-
nication company offering phone plans to consumers. A plan is characterized by a set of features
such as price per month, free minutes in peak hours, free minutes in weekends, free data, price for
additional data, and price per minute to foreign countries. Given two plans xand y, we say that
plan xdominates plan y, if the price per month of xis less than that of y, and xis at least as good
as yin every single feature. In the past, the company offered consumers a certain set of plans St
each month tsuch that no plan in Stis dominated by another plan (in St). The offered plans how-
ever were different each month. Using historical data and assuming that consumers preferences can
be approximated using a multinomial logit, it is possible to perform a robust estimation procedure
to obtain the parameters of such MNL model. Once the parameters are obtained, the assortment
problem consists in finding the best assortment of phones plans Sto maximize the expected rev-
enue. A natural constraint in this problem consisting in enforcing that every phone plan offered
in Scannot be dominated by any other. Section 4 shows that the problem discussed here can
be modelled using the 2SLM and thus solving this problem is reduced to solving an assortment
problem under the 2SLM.
2 Contributions
The first key contribution is to show that the assortment problem can be solved in polynomial time
under the 2SLM. The proof is built upon two unrelated results in optimization: the polynomial-
time solvability of the maximum-independent set in a comparability graph (M¨ohring, 1985) and
a seminal result by Megiddo (1979) that provides an algorithm to solve a class of combinatorial
optimization problems with rational objective functions in polynomial time. This is particularly
appealing since the 2SLM is one of the very few choice models that goes beyond the random utility
model and it allows violations the property known as regularity: the probability of choosing an
alternative cannot increase if the offer set is enlarged. Since many decades ago, there are well-
documented lab experiments where the regularity property is violated (Huber, Payne, and Puto,
1982; Tversky and Simonson, 1993; Herne, 1997).
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The second key contribution is to show that the capacitated assortment problem under the 2SLM
is NP-hard, which contrasts with results on the MNL. We then propose polynomial algorithms for
two interesting subcases of the capacitated assortment problem: (1) When the dominance relation
is attractiveness-correlated and (2) when the transitive reduction of the dominance relation can be
represented as a forest. The proofs use a strong connection between assortments under the 2SLM
and independent sets.
The third and final contribution, is an in-depth study of the pricing problem under the 2SLM.
We first note that changes in prices should be reflected in the dominance relation if the differences
between the resulting attractiveness are large enough. This is formalized by solving the Joint
Assortment and Pricing problem under the Threshold Luce model, where one product dominates
another if the ratio between their attractiveness is bigger than a fixed threshold. Under this
setting, we show that this problem can be solved in polynomial time. The proof relies on the
following interesting facts: (1) An intrinsic utility ordered assortment is optimal; (2) the optimal
prices can be obtained in polynomial time; and (3) it assigns the same price for all products, except
for two of them, the highest and lowest attractiveness ones. Many of these results are extended to
the following cases (1) capacity constrained problems, where the number of products that can be
offered is restricted and (2) position bias, where products are assigned to positions, altering their
perceived attractiveness.
The rest of the paper is organized as follows: Section 3 presents a review of the literature con-
cerning assortment optimization and pricing under variations of the Multinomial Logit. Section 4
formalizes the 2SLM and some of its properties. Section 5 proves that assortment optimization
under the 2SLM is polynomial-time solvable. Section 6 presents the results on the capacitated
version, particularly the NP-hardness of the capacitated version of the problem, but also provide
polynomial time solutions for two special cases. Section 7 present the results for pricing optimisa-
tion under the Threshold Luce model. Section 8 concludes the paper and provides future research
directions. All proofs missing from the main text, are provided in Appendix A.
3 Literature Review
Since the assortment problem and the joint assortment and pricing problem are a very active
research topic, we focus on recent results closely related with this paper and in particular, results
over the multinomial logit model (MNL) (Luce, 1959; McFadden, 1978) and its variants.
Despite the IIA property, the MNL is widely used. Indeed, for many applications, the mean
utility of a product can be modeled as a linear combination of its features. If the features capture
the mean utility associated with each product, then the error between the utilities and their means
may be considered as independent noise and the MNL emerges as a natural candidate for modeling
customer choice. In addition, the MNL parameters can be estimated from customer choice data,
even with limited data (Ford, 1957; Negahban, Oh, and Shah, 2012), because the associated estima-
tion problem has a concave log likelihood function (McFadden, 1974) and it is possible to measure
4
how good the fitted MNL approximates the data (Hausman and McFadden, 1984). Moreover, it is
possible to improve model estimation when the IIA property is likely to be satisfied (Train, 2003).
One of the first positive results on the assortment problem under the multinomial logit model was
obtained by Talluri and Van Ryzin (2004), where the authors showed that the optimal assortment
can be found by greedily by adding products to the offered assortment in the order of decreasing
revenues, thus evaluating at most a linear number of subsets. Rusmevichientong, Shen, and Shmoys
(2010) studied the assortment problem under the MNL but with a capacity constraint limiting the
products that can be offered. Under these conditions, the optimal solution is not necessarily a
revenue-ordered assortment but it can still be found in polynomial time.
Gallego, Ratliff, and Shebalov (2011) proposed a more general attraction model where the
probabilities of choosing a product depend on all the products (not only the offered subset as in
the MNL). This involves a shadow attraction value associated with each product that influence the
choice probabilities when the product is not offered. Davis, Gallego, and Topaloglu (2013) showed
that a slight transformation of the MNL model allows for the solving of the assortment problem
when the choice probabilities follow this more sophisticated attraction model. This continues to
hold when assortments must satisfy a set of totally unimodular constraints.
The Mixed Multinomial Logit (Daly and Zachary, 1978) is an extension of the MNL model,
where different sets of customers follow different MNL models. Under this setting, the problem
becomes NP-hard (Bront, M´endez-D´ıaz, and Vulcano, 2009) and it remains NP-hard even for two
customer types (Rusmevichientong et al., 2014). A branch-and-cut algorithm was proposed by
endez-D´ıaz et al. (2014). Feldman and Topaloglu (2015) proposed methods to obtain good upper
bounds on the optimal revenue. Rusmevichientong and Topaloglu (2012) considered a model where
customers follow a MNL model and the parameters belong to a compact uncertainty set. The firm
wants to hedge against the worst-case scenario and the problem amounts to finding an optimal
assortment under this uncertainty conditions. Surprisingly, when there is no capacity constraint,
the revenue-ordered strategy is optimal in this setting. Jagabathula (2014) proposed a local-search
heuristic for the assortment problem under an arbitrary discrete choice model. Davis, Gallego,
and Topaloglu (2013) and Abeliuk et al. (2016) proposed polynomial time algorithms to solve
the assortment problem under the MNL model with capacity constraint and position bias, where
position bias means that customer choices are affected by the positioning of the products in the
assortment. Recently, Jagabathula and Vulcano (2015) proposed a partial-order model to estimate
individual preferences, where preference over products are modeled using forests. They cluster the
customers in classes, each class being represented with a forest. When facing an assortment S,
customers select, following an MNL model, products that are roots of the forest projected on S.
This approach outperformed state-of-the-art methods when measuring the accuracy of individual
predictions.
Attention has also been devoted to discrete choice models to represent customer choices in
more realistic ways, including models that violate the IIA property (Ben-Akiva and Lerman, 1985).
This property does not always hold in practice (Rieskamp, Busemeyer, and Mellers, 2006), includ-
5
ing when products cannibalize each other (Ben-Akiva and Lerman, 1985). Echenique, Saito, and
Tserenjigmid (2018) identify these violations as perception priorities, and adjust probabilities to
take their effects into account. Gul, Natenzon, and Pesendorfer (2014) provide an axiomatic gen-
eralization of MNL model to address the case where the products share features. Fudenberg and
Strzalecki (2015) propose an axiomatic generalization of a discounted logit model incorporating a
parameter to model the influence of the assortment size.
Customers tend to use rules to simplify decisions, and before making a purchase decision, they
often narrow down the set of alternatives to chose from, using different heuristics to make the
decision process simpler. Several models of consider-then-choose models have been proposed in
the literature, related with attention filters, search costs, feature filters, among others, another
reasonable way to discard options, is when the difference between attractiveness is so evident, that
the less attractive alternative, even when it is offered, is never picked (as in the Threshold Luce
model, Echenique and Saito (2018)). Any of the heuristics mentioned before allows the consumer
to restrict her attention to a smaller set usually referred in the literature as consideration set. This
effect also provokes that offered product might result having zero-probability choices.
Several models have been proposed to address the issue of zero-probability choices. Masatlioglu,
Nakajima, and Ozbay (2012) propose a theoretical foundation for maximizing a single preference
under limited attention, i.e., when customers select among the alternatives that they pay attention
to. Manzini and Mariotti (2014) incorporate the role of attention into stochastic choice, proposing a
model in which customers consider each offered alternative with a probability and choose the alter-
native maximizing a preference relation within the considered alternatives. This was axiomatized
and generalized in Brady and Rehbeck (2016), by introducing the concept of random conditional
choice set rule, which captures correlations in the availability of alternatives. This concept also
provided a natural way to model substitutability and complementarity.
Payne (1976) showed that a considerable portion of the subjects in his experimental setting
use a decision process involving a consideration set. Numerous studies in marketing also validated
a consider-then-choose decision process. In his seminal work Hauser (1978) observed that most
of the heterogeneity in consumer choice can be explained by consideration sets. He shows that
nearly 80% of the heterogeneity in choice is captured by a richer model based in the combination of
consideration sets and logit-based rankings. The rationale behind this observation is that first stage
filters eliminate a large fraction of alternatives, thus the resulting consideration sets are composed
of a few products in most of the studied categories (Belonax Jr and Mittelstaedt, 1978; Hauser and
Wernerfelt, 1990). Pras and Summers (1975) and Gilbride and Allenby (2004) empirically showed
that consumers form their consideration sets by a conjunction of elimination rules. Furthermore,
there are empirical results showing that a Two-Stage model including consideration sets better
fits consumer search patterns than sequential models (De los Santos, Horta¸csu, and Wildenbeest,
2012).
Form a customer standpoint, the use for consider-then-choose models alleviate the cognitive
burden of deciding when facing too many alternatives Tversky (1972a,b); Tversky and Kahneman
6
(1974); Payne, Bettman, and Luce (1996). When dealing with a decision under limited time and
knowledge, customers often recur to screening heuristics as show in Gigerenzer and Goldstein (1996).
Psychologically speaking, customers as decision makers need to carefully balance search efforts and
opportunity costs with potential gains, and consideration sets help to achieve that goal (Roberts and
Lattin, 1991; Hauser and Wernerfelt, 1990; Payne, Bettman, and Luce, 1996). Recently Jagabathula
and Rusmevichientong (2017) proposed a Two-Stage model where customers consider only the
products are contained within certain range of their willingness to pay. Aouad, Farias, and Levi
(2015) explored consider-then-choose models where each costumer has a consideration set, and a
ranking of the products within it. The customer then selects the higher ranked product offered.
The authors studied the assortment problem under several consideration sets and ranking structure,
and provide a dynamic programming approach capable of returning the optimal assortment in
polynomial time for families of consideration set functions originated by screening rules Hauser,
Ding, and Gaskin (2009). Dai et al. (2014) considered a revenue management model where an
upcoming customer might discard one offered itinerary alternative due to individual restrictions,
such as time of departure. Wang and Sahin (2018) studied a choice model that incorporates product
search costs, so the set that a customer considers might differ from what is being offered.
Multi-product price optimisation under the MNL and the NL has been studied since the models
were introduced in the literature. One of the first results on the structure of the problem is due
to Hanson and Martin (1996), where they show that the profit function for a company selling
substitutable products when customers follow the MNL model is not jointly concave in price. To
overcome this issue, in Song and Xue (2007) and later in Dong, Kouvelis, and Tian (2009), the
authors show that even when the profit function is not concave in prices, it is concave in the market
share and there is a one-to-one correspondence between price and market share. Multiple studies
shown that under the MNL where all products share the same price sensitivity parameter, the
mark-up which is simply the difference between price and cost, remains constant for all products
at optimality (Anderson, Depalma, and Thisse, 1992; Hopp and Xu, 2005; Gallego and Stefanescu,
2009; Besbes and Saur´e, 2016). Furthermore, the profit function is also uni-modal on this constant
quantity and it has a unique optimal solution, which can be determined by studying the first order
conditions.
Li and Huh (2011) showed the same result for the NL model. Up to that point, all previous
results assumed an identical price sensitivity parameter for all products. Under the MNL, there
is empirical evidence that shows the importance of allowing different price sensitivity parameters
for each product (Berry, Levinsohn, and Pakes, 1995; Erdem, Swait, and Louviere, 2002). There
is is also evidence in B¨orsch-Supan (1990) that restricting the nest specific parameters to the unit
interval results in rejection of the NL model when fitting the data, thus recommending to relax
this assumption. The problem when relaxing this condition, is that the profit function is no longer
concave on the market share, which complicates the optimization task. In Gallego and Wang
(2014) the authors considered a NL model with differentiated price sensitivities, and found that
the adjusted mark-up, defined as price minus cost minus the reciprocal of the price sensitivity is
7
constant for all products within a nest at optimality. Furthermore, each nest also has an adjusted
next-level markup which is also invariant across nests, which reduces the original problem to a
one variable optimization problem. Additional theoretical development can be found in Rayfield,
Rusmevichientong, and Topaloglu (2015); Kouvelis, Xiao, and Yang (2015) but there are restricted
to the Two-Stage nested logit model. In Huh and Li (2015) some of the results were extended to a
multi-stage nested logit model for specific settings, but also show that the equal mark-up property
fails to hold in general for products that do not share the same immediate parent node in the
nested choice structure, even when considering identical price sensitivity parameters. Li and Huh
(2011) and Gallego and Wang (2014) extend to the multi-stage NL model and show that an optimal
pricing solution can still be found by means of maximizing a scalar function.
There are some interesting results for other models that share similarities with the MNL, and
therefore are closely related with the model that we are studying. In Wang and Sahin (2018), the
authors incorporate search cost into consumer choice model. The results on this paper for the Joint
Assortment and Pricing are similar to the ones that we study in Section 7, in that many structural
results that holds at optimality for their model, are also satisfied in our studied case. They show
that the quasi-same price policy (that charges the same price for all products but one, the least
attractive one) was optimal for this model. Interestingly, the Joint Assortment and Pricing results
under the Threshold Luce Model has a slightly different result: The optimal pricing is a fixed price
for all products, except for the most attractive and least attractive ones. This led to a situation
where there are many possible prices, not just two.
Recently Alptekino˘glu and Semple (2016) hast studied in depth a model which was originally
due to Daganzo (1979) that assumes a negatively skewed distribution of consumer utilities. The
resulting choice probabilities have an interesting consequence in the optimal pricing policy: They
allow for variable mark-ups in optimal prices that increase with expected utilities.
The model considered in this paper is a variant of the MNL, proposed by Echenique and Saito
(2018) and called the Two-Stage Luce model ; It handles zero-probability choice by introducing the
concept of dominance, meaning that if a product xdominates a product y, then yis never selected
in presence of y. And therefore the consideration set is formed by considering only non-dominated
products in the offered assortment, allowing flexibility on the consideration set formation due to
the nature of the dominance relation. Once the consideration set is formed, the customer choose
according to an MNL on the remaining alternatives. In the following section we describe this model
in detail, and show some examples that highlight many practical applications for it.
4 The Two-Stage Luce model
The 2SLM (Echenique and Saito, 2018) overcomes a key limitation of the MNL: The fact that
a product must have zero attractiveness if it has zero probability to be chosen in a particular
assortment. This limitation means that the product cannot be chosen with positive probability
in any other assortment. The 2SLM eliminates this pathological situation through the concept of
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consideration function which, given a set of products S, returns a subset of Swhere each product
has a positive probability of being selected. Let Xdenotes the set of all products and let a(x)>0
be the attractiveness of product xX. For notational convenience, we use axto denote the
attractiveness of product x, i.e., ax=a(x). We extend the attractiveness function to consider the
outside option, with index 0 and a0=a(0) 0, to model the fact that customers may not select
any product. As a result, the attractiveness function has signature a:X∪ {0} → R+. Given an
assortment AX, a stochastic choice function ρreturns a probability distribution over A, i.e.,
ρ(x, A) is the probability of picking xin the assortment A. The 2SLM is a sub case of the general
Luce model presented in Echenique and Saito (2018), and independently discovered in Ahumada
and ¨
Ulk¨u (2018), which is defined below.
Definition 1 (General Luce Function , Echenique and Saito (2018)).A stochastic choice function
ρis called a general Luce function if there exists an attractiveness function a∪ {0}:XR+and
a function c: 2X\ ∅ → 2X\ ∅ with c(A)Afor all AXsuch that
ρ(x, A) =
ax
Pyc(A)ay+a0if xc(A),
0 if x /A.
(1)
for all AX. We call the pair (a, c) a general Luce model.
The function c(which is arbitrary) provides a way to capture the support of the stochastic choice
function ρ. As observed in Echenique and Saito (2018), there are two interesting cases worthy of
being mentioned:
1. If c(S) is a singleton for all SX, then ρ(x, S) is a deterministic choice.
2. If c(S) = Sfor all SX, then the 2SLM coincides with the MNL.
Two special cases of this model were provided in Echenique and Saito (2018). The first is
the two-stage Luce model. This model restricts c, such that the c(A) represents the set of all
undominated alternatives in A.
Definition 2 (two-Stage Luce model (2SLM), Echenique and Saito (2018)).A general Luce model
(a, c) is called a 2SLM if there exists a strict partial order (i.e. transitive, antisymmetric and
irreflexive binary relation) such that:
c(A) = {xA| 6 ∃yA:yx}.(2)
We call dominance relation.
As a result, any 2SLM can be described by an irreflexive, transitive, and antisymmetric relation
that fully captures the relation between products. The second model presented in Echenique and
Saito (2018), which is a particular case of the 2SLM, is the Threshold Luce Model (TLM), where
The definition is slightly different: It makes the outside option effect a0explicit in the denominator.
9
they explain dominance in terms of how big the attractiveness are when compared with each other,
so cis strongly tied to a. More specifically, for a given threshold t > 0, the consideration set c(S)
for a set SXis defined as:
c(S) = {yS| 6 ∃xS:ax>(1 + t)ay}.(3)
In other words, xyif and only if ax
ay>(1 + t). Intuitively, an attractiveness ratio of more than
(1 + t) means that the less-preferred alternative is dominated by the more-preferred alternative.
Observe that the relation is clearly irreflexive, transitive, and antisymmetric.
The dominance relation can thus be represented as a Directed Acyclic Graph (DAG), where
nodes represent the products and there is a directed edge (x, y) if and only if xy. Sets satisfying
c(S) = Sare anti-chains in the DAG, meaning that there are no arcs connecting them. For
instance, consider the Threshold Luce model defined over X={1,2,3,4,5}with attractiveness
values a1= 12, a2= 8, a3= 6, a4= 3 and a5= 2, and threshold t= 0.4. We have that ijiff
ai>1.4aj.
The DAG representing this dominance relation is depicted in Figure 1.
1
a1= 12
2
a2= 8
3
a3= 6
4
a4= 3
5
a5= 2
Figure 1: Example of a DAG for a General Threshold Luce model
In the following example, we show that the 2SLM admits regularity violations, meaning that it
is possible that the probability of choosing a product can increase when we enlarge the offered set.
Since regularity is satisfied by any choice model based on random utility (RUM), this shows that
the 2SLM is not contained in the RUM class .
Example 1. Consider the following instance of the Threshold Luce model (which is a special case
of the 2SLM). Let X={1,2,3,4}with attractiveness a1= 5, a2= 4, a3= 3 and a4= 3. Consider
t= 0.4 and the attractiveness of the outside option a0= 1. For the offer set {2,3,4}, the probability
of selecting product 2 is 4/11 since no product dominates each other. However, if we add product 1
to the offer set, i.e. if we offer all four products, then the probability of selecting product 2 increases
to 4/10, because products 3 and 4 are now dominated by product 1.
The Two-Stage Luce Model allows to accommodate different decision heuristics and market
scenarios by specifying the dominance relation responding to a specific set of rules. Two cases
where this can be observed are provided below.
Observe that this implies that the 2SLM is not contained by the Markov chain model proposed by (Blanchet,
Gallego, and Goyal, 2016) since this last one belongs to the RUM class (Berbeglia, 2016).
10
Feature Difference Threshold: Assume that each product has a set of features F={1, . . . , m}.
A product xcan then be represented by a m-dimensional vector xRm. Assume that the
perceived relevance of each feature kis measured by a weight νk, so that the utility perceived by
the customers can be expressed as a weighted combination of their features u(x) = Pm
k=1 νk·xk.
The dominance relation can be defined as xyu(x)u(y) = Pm
k=1 νk(xkyk)T,
where T > 0 is a tolerance parameter that represents how much difference a customer allows before
considering that an alternative dominates another. The dominance relation is irreflexive, transitive,
and antisymmetric and hence it can be used to define an instance of the 2SLM. One can easily
show that this model is a special case of the TLM.
Price levels: Suppose we have Nproducts, each product ihas kiprice levels. Let xil be product
iwith price pil attached and it corresponding attractiveness ail, we assume that for each product
iprices pik satisfy pi1< pi2< . . . , piki. Naturally, xi1xi2. . . xiki, because for the same
product the customer is going to select the one with the lowest price available. Each price level
for each product can still dominate or be dominated by other products as well, as long as the
dominance relation is irreflexive, transitive and antisymmetric. This setting can be modelled by
the Two-Stage Luce model in a natural way.
5 Assortment Problems Under the Two-Stage Luce model
This section studies the assortment problem for the 2SLM using the definitions and notations
presented earlier. Let r:X∪ {0} → R+be a revenue function associated with each product and
satisfying r(0) = 0. The expected revenue of a set SXis given by
R(S) = X
ic(S)
ρ(i, S)r(i).(4)
The assortment problem amounts to finding a set
Sargmax
SX
R(S)
yielding an optimal revenue of
R= max
SXR(S).
Observe that every subset SXcan be uniquely represented by a binary vector x∈ {0,1}nsuch
that iSif and only if xi= 1. Using this bijection, the search space for Scan be restricted to
D={x∈ {0,1}n| ∀st:xs+xt1}
where Drepresents all the subsets satisfying S=c(S), which means that no product on Sdominates
another product in S. There is always an optimal solution Sthat belongs to Dbecause R(S) =
R(c(S)) and c(S)Dfor all sets Sin X. As a result, the Assortment Problem under the 2SLM
11
(AP-2SLM) can be formulated as
maximize
xPn
i=1 riaixi
Pn
i=1 aixi+a0
subject to x∈ D
(AP-2SLM)
where riand airepresent r(i) and a(i) for simplicity.
An effective strategy for solving many assortment problems consists in considering revenue-
ordered assortments, which are obtained by choosing a threshold ρand selecting all the products
with revenue at least ρ. This strategy leads to an optimal algorithm for the assortment problem
under the MNL. Unfortunately, it fails under the 2SLM because adding a highly attractive product
may remove many dominated products whose revenues and utilities would lead to a higher revenue.
Example 2 (Sub-Optimality of Revenue-Ordered Assortments).Consider a Threshold Luce model
with X={1,2,3}, revenues r1= 88, r2= 47, r3= 46, attractiveness a0= 55, a1= 13, a2= 26, a3=
15 and t= 0.6. Then xyiff ax>1.6aywhich gives 2 1 and 2 3. Consider the sets SX
satisfying S=c(S):
S R(S)
{1}16.824
{2}15.086
{3}9.857
{1,3}22.096
The optimal revenue is given by assortment {1,3}, while the best revenue-ordered assortment
under the 2SLM is S={1}, yielding almost 24% less revenue.
To solve problem AP-2SLM, consider first the MaxAtt problem defined over the same set of
constraints. Given weights ciR(1 in), the MaxAtt problem is defined as follows:
maximize
x
n
X
i=1
cixi
subject to x∈ D
(MaxAtt)
We now show that (MaxAtt) can be reduced to the maximum weighted independent set problem
in a directed acyclic graph with positive vertex weights. An independent set is a set of vertices I
such that there is no edge connecting any two vertices in I. The maximum weighted independent
set problem (MWIS) can be stated as follows:
Definition 3. Maximum Weighted Independent Set Problem: Given a graph G= (V, E ) with a
weight function w:VR, find an independent set IargmaxI∈I PiIw(i), where Iis the set
of all independent sets.
12
Recall that the dominance relation can be represented as a DAG Gwhich includes an arc (u, v )
whenever uv. As a result, the condition x∈ D implies that any feasible solution to (MaxAtt)
represents an independent set in Gand maximizing Pn
i=1 cixiamounts to finding the independent
set maximizing the sum of the weights. Since the dominance relation is a partial order, the DAG
representing the dominance relation is a comparability graph. The following result is particularly
useful.
Theorem 1 (M¨ohring (1985)).The maximum weighted independent set is polynomially-solvable
for comparability graphs with positive weights.
We are ready to present our first result.
Lemma 1. (MaxAtt)is polynomial-time solvable.
Proof. We first show that we can ignore those products with a negative weight. Let ˆ
X={iX|
ci>0}and ˆ
D={x∈ {0,1}n| ∀s, t ˆ
X, s t:xs+xt1}. Solving (MaxAtt) is equivalent to
solving:
maximize
xX
iˆ
X
cixi
subject to xˆ
D
(Reduced MaxAtt)
Indeed, consider an optimal solution xto Problem MaxAtt and assume that there exists iX
such that ci<0 and x
i= 1. Define ˆxlike xbut with ˆxi= 0. ˆxhas a strictly greater value for
the objective function in Reduced MaxAtt than xhas, and is feasible since setting a component
to zero cannot violate any constraint (i.e., ˆx∈ D). This contradicts the optimality of x. Now
Problem Reduced MaxAtt can be reduced to solving an instance of Problem MWIS in a DAG with
positive weights that corresponds to the dominance relation. This DAG is a comparability graph
and the result follows from Theorem 1.
The next step in solving the assortment problem under the 2SLM relies on a result by Megiddo
Megiddo (1979). Let Dbe a domain defined by some set of constraints and consider Problem A
maximize
x
n
X
i=1
cixi
subject to xD
(A)
and its associated Problem B:
maximize
x
a0+Pn
i=1 aixi
b0+Pn
i=1 bixi
subject to xD.
(B)
Using this notation, Megiddo’s theorem can be stated as follows.
13
Theorem 2 (Megiddo (1979)).If Problem Ais solvable within O(p(n)) comparisons and O(q(n))
additions, then Problem Bis solvable in O(p(n)(q(n) + p(n))) time.
We are now in position to state our main theorem of this section.
Theorem 3. The assortment problem under the Two-Stage Luce model is polynomial-time solvable.
Proof. Recall that the assortment problem under the 2SLM (AP-2SLM) can be formulated as
maximize
xPn
i=1 riaixi
Pn
i=1 aixi+a0
subject to x∈ D
(5)
where D={x∈ {0,1}n| ∀st:xs+xt1}.
The problem of maximizing the numerator in (5) is exactly the MaxAtt problem. By Lemma
1, this is polynomial-time solvable. Now observe that (5) (i.e., problem AP-2SLM) can be seen as
a Problem B. Therefore, by Theorem 2, the assortment problem under the 2SLM is solvable in
polynomial time.
In addition to solving the assortment problem under the 2SLM, Theorem 3 is interesting in that it
solves the assortment problem under a Multinomial Logit with a specific class of constraints. It can
be contrasted with the results by Davis, Gallego, and Topaloglu (2013), where feasible assortments
satisfy a set of totally unimodular constraints. They show that the resulting problem can be solved
as a linear program. However, the 2SLM introduces constraints that are not necessarily totally
unimodular as we now show.
Example 3. Consider X={1,2,3,4}and 1 3,14,23,24,and 3 4. The constraint
matrix that defines the feasible space (D) for this instance is:
M=
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 1
0 0 1 1
where each row represents a constraint xu+xv1. meaning that just one end of the edge can be
selected at the time. Camion (1965) proved that Mis totally unimodular if and only if, for every
(square) Eulerian submatrix A of M,Pi,j aij 0 (mod 4). Consider the sub-matrix corresponding
to the first, second, and fifth rows and the first, third, and fourth columns
N=
110
101
011
14
Matrix Nis eulerian (The sums of every element on each row or on each column is a multiple of
2). But the sum of all elements of Nis 6 6≡ 0 (mod 4) and hence Mis not totally unimodular.
We close this section by explaining how our results can be extended to a more general setting.
Gallego, Ratliff, and Shebalov (2014) proposed the general attraction model (GAM) to describe
customer behaviour, that alleviates some deficiencies of the MNL. More specifically, the intuition
behind this choice model is that whenever a product is not offered, then its absence can potentially
increase the probability of the no-purchase alternative, as consumers can potentially look for the
product elsewhere, or at a later time. To achieve this effect, for each product jthe model considers
two different weights: vjand wj, usually with 0 wjvj. If product jis offered, then its
preference weight is vj. But if jis not offered, then the preference weight of the outside option is
increased by wj. For all jX, let evj=vjwjand ˜v0=v0+PkXwk. Using this notation, the
probabilities associated with the GAM model can be recovered by means of the following equation:
ρ(j, S) =
vj
PiSevj+ev0if jS,
0 if j /S.
(6)
Observe that the resulting assortment problem will has the same functional form than problem
AP-2SLM, with a slight modification on the coefficients in the denominator. Thus, we can apply the
same solution technique described in Theorem 3 to find the optimal assortment for the GAM.
6 The Capacitated Assortment Problem
In many applications, the number of products in an assortment is limited, giving rise to capacitated
assortment problems. Let C(1 Cn) be the maximum number of products allowed in an
assortment. The Capacitated Assortment Problem under the Two-Stage Luce Model (C2SLMAP) is
given by
maximize
xPn
i=1 riaixi
Pn
i=1 aixi+a0
subject to x∈ DC
(C2SLMAP)
where DC={x∈ {0,1}n| ∀(s, t)∈ R xs+xt1Pn
i=1 xiC}. As before, it is useful to define
its capacitated maximum-attractiveness counterpart (C-MaxAtt), i.e.,
maximize
x
n
X
i=1
cixi
subject to x∈ DC
(C-MaxAtt)
This section first proves that the capacitated assortment problem under the 2SLM is NP-hard. The
reduction uses the Maximum Weighted Budgeted Independent Set (MWBIS) problem proposed by
Bandyapadhyay (2014) which amounts to finding a maximum weighted independent set of size not
greater than C. Kalra et al. (2017) showed that Problem (MWBIS) is NP-hard for bipartite graphs.
15
Theorem 4. Problem (C2SLMAP)is NP-hard (under Turing reductions).
It is interesting to mention that Problem (C-MaxAtt) is equivalent to finding an anti-chain of
maximum weight among those of cardinality at most C. This problem (MWLA) was proposed by
Shum and Trotter (1996) and its complexity was left open, but the above results show that it is
also NP-hard. Bandyapadhyay (2014) studied Problem (MWBIS) for various types of graphs (e.g.,
trees and forests), but the dominance relation of the 2SLM can never be a tree since it is transitive
(unless we consider a graph with a single vertex).
In light of this NP-hardness result, the rest of this section presents polynomial-time algorithms
for two special cases of the dominance relation.
6.1 The Two-Stage Luce model over Tree-Induced Dominance Relations
Let Rbe the transitive reduction of the irreflexive, antisymmetric, and transitive relation . This
section considers the capacitated assortment problem when the relation Rcan be represented as
a tree. Without loss of generality, we can assume that the tree contains all products. Otherwise, we
can add another product with zero weight that dominates all original products. This new product
will be the root of the tree and the products not in the original tree will be the children of the root.
Similarly, the same transformation applies to the case when Ris a forest. Here all the trees in
the forest will be children of the new product.
We show how to solve Problem (C-MaxAtt). The result follows again by applying Megiddo’s
theorem. The first step of the algorithm simply removes all products with negative weight: Their
children can be added to the parent of the deleted vertex. The main step then solves (C-MaxAtt)
bottom-up using dynamic programming from the leaves. For simplicity, we present the recurrence
relations to compute the weight of the optimal assortment. It is easy to recover the optimal
assortment itself. The recurrence relations compute two functions:
1. A(k, c) which returns the weight of an optimal assortment using product kand its descendants
in the tree representation of Rfor a capacity c;
2. A+(S, c) which, given a set Sof vertices that are children of a vertex k, returns the weight
of an optimal assortment using the products in Sand their descendants for a capacity c.
The key intuition behind the recurrence is as follows. If vis a vertex and v1and v2are two of
its children, v1does not dominate v2or any of its descendants. Hence, it suffices to compute the
best assortments producing A(v1,0),...,A(v1, C ) and A(v2,0),...,A(v2, C) and to combine them
optimally. The recurrence relations are defined as follows (vXand 1 cC):
A(v, 0) = 0;
A(v, c) = max(cv,A+(children(v), c));
16
and
A+(,0) = 0;
A+(S, c) = max
n1,n20
n1+n2=c
A+(S\ {e}, n1) + A(e, n2) where e= argmax
iS
ci.
where children(p) denotes the children of product pin the tree. Note that A+(S, c) is computed
recursively to obtain the best assortment from the products in Sand their descendants. Using
these recurrence relation, the following Theorem follows:
Theorem 5. Let a dominance relation whose relation Ris a tree containing all products. The
capacitated assortment problem under the 2SLM and is polynomial-time solvable.
6.2 The Attractiveness-Correlated Two-Stage Luce model
The second special case considers a dominance relation that is correlated with attractiveness.
Definition 4 (Attractiveness-Correlated Two-Stage Luce model).A Two-Stage Luce model is
attractiveness-correlated if the dominance relation satisfies the following two conditions:
1. If xy, then ax> ay.
2. If xyand az> ax, then zy.
The first condition simply expresses that product xcan only dominate product yif the attractive-
ness of xis greater than the attractiveness of y. The second condition ensures that, if xdominates
y, then any product whose attractiveness is greater than xalso dominates y. The induced domi-
nance relation is irreflexive, anti-symmetric, and transitive. A particular case of this model, is the
Threshold Luce model.
When customers follow the Threshold Luce model, they form their consideration sets based on
the attractiveness of products. Without loss of generality, we can assume a1a2. . . an,
unless stated otherwise. For a set S, the associated consideration set c(S) may be a proper subset
of S, but for the purpose of assortment optimization, we don’t have incentives to offer sets including
products that are not even consider by customers, so we can restrict our search for optimal solutions
to sets where c(S) = S. A necessary and sufficient condition for this to happen is maxiSai
miniSai1 + t.
Meaning that largest ratio between attractiveness is not greater than 1+t, so no dominance relation
appears.
The firm now needs to carefully balance the inclusion of high-attractiveness products and their
prices to maximize the revenue. In the following example we show that revenue ordered assortments
are not optimal under the Threshold Luce Model. In fact, this strategy can be arbitrarily bad.
Example 4 (Revenue ordered assortments are not optimal).Consider the following product con-
figuration. Let N+ 1 products, with prices p1for the first product, and αp1for the rest of them,
with α < 1. The attractiveness for all products is a1for the first product and γa1for all the rest,
17
such as in the presence of product 1, all the rest of the products are ignored. To complete the
set up, let a0the attractiveness of the outside option. The best revenue ordered assortment is to
consider product 1, given a revenue of:
R0=R({1}) = p1a1
a1+a0
But, if Nis big enough (at least bigger than 1
αγ ), is more profitable to show SN=X\ {1},
resulting in a revenue of:
R=R(SN) = N·αp1γa1
N·αγa1+a0
Now, if we calculate the ratio if this two values, R0and Rand let Ntend to infinity we have:
R0
R= lim
N→∞
p1a1
a1+a0
N·αp1γa1
N·αγa1+a0
R0
R= lim
N→∞
p1a1
a1+a0
·N·αγa1+a0
N·αp1γa1
R0
R=a1
a1+a0
(7)
Observe that this last expression is the market share of offering just product 1, which can
be arbitrarily bad by either making a1as small as desired, or making the outside option more
attractive.
The capacitated assortment optimization can be solved in polynomial time under the Attractiveness-
Correlated Two-Stage Luce model. Consider an assortment whose product with the largest attrac-
tiveness is k. This assortment cannot contain any product dominated by k. Moreover, if k1and k2
are two other products in this assortment, then k1cannot dominate k2since kwould also dominate
k2. As a result, consider the set
Xk={iX|aiak&k6 i}.
No product in Xkdominates any other product in Xkand hence the C2SLMAP reduces to a tradi-
tional assortment problem under the MNL. This idea is formalized in Algorithm 1, where CMLMAP
is a traditional algorithm for the MNL. The algorithm considers each product in turn and the
products that it does not dominate and applies a traditional capacitated assortment optimization
under the MNL. The best such assortment is the solution to the capacitated assortment under the
attractiveness-correlated 2SLM.
18
Algorithm 1: Capacitated Assortment Optimization under the Attractiveness-Correlated
2SLM.
Data: X, , r, a
Result: Optimal Assortment S
R(S)=0for k= 1, . . . , n do
Xk={iX|aiak&k6 i}
Sk=CMLMAP(Xk, r, a)
if R(Sk)> R(S)then
S=Sk
end
end
return S
Theorem 6. C2SLMAP can be solved in polynomial time for Attractiveness-Correlated instances.
Proof. To show correctness, it suffices to show that the optimal assortment must be a subset of
one of the Xk(1 kn). Let Abe the optimal assortment and assume that kis its product
with the largest attractiveness (break ties randomly). Amust be included in Xksince otherwise
it would contain a product xsuch that kx(contradicting feasibility) or such that a(x)>
a(k) (contradicting our hypothesis). The correctness then follows since there is no dominance
relationship between any two elements in each of Xk. The claim of polynomial-time solvability
follows from the availability of polynomial-time algorithms for the assortment problem under the
MNL and the fact that are exactly ncalls to such an algorithm.
7 Joint Assortment and Pricing under the Threshold Luce model
The previous sections provides solutions to the Assortment Optimization problem under the Two-
Stage Luce model. This section aims at determining how to assign prices to products in order to
maximise the expected revenue. It studies the Joint Assortment and Pricing Problem under the
Threshold Luce model, by making the attractiveness of each product dependent upon its price.
Let p= (p1, . . . , pn) be the price vector, where such that piR+∪ {∞} represents the price of
product i. Since the price will affect the attractiveness aiof product i, the presentation makes this
dependency explicit by writing ai(pi) whose form in this paper is specified by
ai(pi) = exp(uipi) (8)
where uiis the intrinsic utility of product iand the value vi=uipiis called the net utility of
product i. Assigning an infinite price to a product is equivalent to not offering the product, as the
attractiveness, and therefore the probability of selecting the product, becomes 0. Without loss of
generality, products are indexed in a decreasing order by intrinsic utility.
19
i uipiai(pi)p0
iai(p0
i)
1 ln(10) ln(3) 3.3 ln(4) 2.5
2 ln(8) ln(3) 2.6 ln(4) 2
3 ln(6) ln(3) 2 ln(3) 2
4 ln(3) ln(3) 1 ln(2) 1.5
Table 1: Summary of utilities, prices and attractiveness for the two proposed scenarios.
1
a1(p1)=3.3
2
a2(p2)=2.6
3
a3(p3)=2
4
a4(p4) = 1
Figure 2: The DAG for the first scenario where all prices are fixed to ln(3) and the threshold is
t= 0.5. Product 1 dominates products 3 and 4, and product 2 dominates product 4.
1
a1(p0
1)=2.5
2
a2(p0
2) = 2
3
a3(p0
3)=2
4
a4(p0
4)=1.5
Figure 3: The DAG for the second scenario where all prices are fixed to (ln(4),ln(4),ln(3),ln(2))b
and the threshold is t= 0.5. Only product 1 dominates product 4.
The following definition is an extension of the definition of a consideration set given an assort-
ment Swhen each product ihas a price pi.
Definition 5. Given an assortment S, a price vector p= (p1, p2, . . . , pn) and a threshold t, the
consideration set c(S, p) for the Threshold Luce model is defined as:
c(S, p) = {jS| 6 ∃iS:ai(pi)>(1 + t)aj(pj)}.(9)
The influence of the price vector over the dominance relations is given by the following example:
Example 5. [Price effect on the dominance relation] Consider the Threshold Luce model defined
over X={1,2,3,4}with utilities u1= ln(10), u2= ln(8), u3= ln(6) and u4= ln(3), and consider
first a scenario where all products have the same price pi= ln(3) i= 1,...,4. Consider also
a second scenario with prices equal to p0
1= ln(4), p0
2= ln(4), p0
3= ln(3) and p0
4= ln(2). For a
threshold t= 0.5, we have that ijiff ai(pi)>1.5aj(pj). A table summarizing the utilities, prices,
and attractiveness for both scenarios is given in Table 1 and the DAGs depicting the dominance
relations for the two scenarios are given in Figures 2 and 3.
20
It is also necessary to update the definition of ρin Definition 1, since it now depends on the
price of all products in the assortment. The definition of ρ:X∪ {0} × 2X×(R+∪ ∞)n[0,1]
becomes:
ρ(i, S, p) =
ai(pi)
Pjc(S,p)aj(pj)+a0,if ic(S, p),
0 if i /c(S, p).
(10)
where a0is the attractiveness of the outside option.
The expected revenue (ER) of an assortment SXand a price vector pRn
+is given by
R(S, p) = X
ic(S,p)
ρ(i, S, p)pi.(ER)
A pair (S, p) with SXand p(R+∪ ∞)nis valid if S={i:pi<∞} and c(S, p) = S. Let
Vbe the set of all valid pairs (S, p). Observe that one can always restrict the search for optimal
solutions to V. Indeed, all dominated products can be given an infinite price and removing them
from the original assortment yields the exact same revenue.
The Joint Assortment and Pricing problem aims at finding a set Sand a price vector p
satisfying
(S, p)argmax
(S,p)∈V
R(S, p)
and yielding an optimal revenue of
R=R(S, p).
First observe that the strategy used to solve this problem under the multinomial logit does not
carry over to the Threshold Luce Model. Under the multinomial logit, the optimal solution for
the joint assortment and pricing problem is a fixed adjusted margin policy (Wang, 2012) which,
for equal price sensitivities and normalised costs, translates to a fixed price policy. As shown in Li
and Huh (2011), the optimal solution for the pricing problem under the multinomial logit can be
expressed in closed form using the Lambert function W(x) : [0,)[0,) which is defined as
the unique function satisfying:
x=W(x)eW(x)x[0,).(11)
Using this function, the optimal revenue can be expressed as:
R=WPiXexp(ui1)
a0(12)
The prices are all equal and satisfy: pi= 1 + RiX. The following example shows that
fixed-price policy is not optimal under the Threshold Luce Model.
Example 6 (Fixed-Price policy is not optimal).Consider 11 products with product 1 having utility
u= 2 and all remaining 10 products having utility u0= 1. Consider a0= 1 and t= 1. Observe
that, for any fixed price, product 1 always dominates the other 10 products having lower utility, as
21
exp(uu0) = exp(1) = e > (1 + t) = 2. Therefore, the optimal revenue for a a fixed price strategy
is:
Rfixed =Wexp(u1)
a0=W(e)=1.
As a result, the 10 lower utility products are completely ignored and only product 1 contributes to
the revenue.
Consider the following price scheme now: let the price for product 1 be p= 1.8 and let the
price be p0= 1.4 for the remaining products. Product 1 does not dominate any other product now.
Indeed, for any 1 < k 11,
a1
ak
= exp((up)(u0p0)) = exp((2 1.8) (1 1.4)) 1.822 <1 + t= 2,
which yields a revenue of:
R0=p·exp(up) + 10 ·p0exp(u0p0)
exp(up) + 10 ·exp(u0p0) + a0
=1.8·exp(2 1.8) + 10 ·1.4 exp(1 1.4)
exp(2 1.8) + 10 ·exp(1 1.4) + 1 1.298,
This pricing scheme improves upon the fixed-price policy, yielding a revenue almost %30 higher.
The intuition behind this example is as follows: For a fixed price strategy, the only factor
affecting dominance is the intrinsic utilities because the prices vanish when calculating the ratio
between two attractiveness. This means that the solution can potentially miss the benefits of low
attractiveness products which are dominated by the most attractive product.
It is thus important to understand the structure of an optimal solution for the Joint Assortment
and Pricing problem under the Threshold Luce model. The first result states that, for any optimal
solution (S, p), all product prices are greater or equal than R, where Rdenotes the revenue
achieved at optimality.
Proposition 1. In any optimal solution (S, p), for all iS,p
iR.
The proof is by contradiction: Removing products with a price lower than Ryields a greater
revenue. The next proposition characterises the optimal assortment of products of any optimal
solution to the Joint Assortment and Pricing problem. Recall that the products are indexed by
decreasing utility ui. Thus, the set of products [k] := {1, . . . , k}, (with 0 < k n) is said to be an
intrinsic utility ordered set. The following proposition holds:
Proposition 2. Let (S, p)denote an optimal solution. Then S= [k]for some kn.
The following Lemma due to Wang and Sahin (2018) is useful to prove some of the upcoming
propositions. For completeness, its proof is also in Appendix A.
Lemma 2 (Lemma 1, Wang and Sahin (2018)).Let H(pi, pj) := pi·exp(uipi) + pj·exp(ujpj),
where exp(uipi) + exp(ujpj) = T. Then, H(pi, pj)is strictly unimodal with respect to pior
pj, and it achieves the maximum at the following point:
p
i=p
j= ln ((exp(ui) + exp(uj))/T ) (13)
22
Observe that setting the price of a product to is equivalent to not showing it to consumers.
By Proposition 2, one can always find an optimal solution that is intrinsic utility ordered. Given a
price vector pRn, let γ(p) : Rn[n] be defined as γ(p).
=maxi[n]is.t pi<. Intuitively,
this is the last non-infinite price. Proposition 3 shows that, at optimality, the finite prices are
non-increasing in i, meaning that lower prices are assigned to lower utility products.
Proposition 3. The prices at an optimal solution (S, p)satisfy p
ip
i+1 i[γ(p)1].
Moreover, if i, j Ssatisfy ui=uj, then p
i=p
j.
Recall that the net utility of product iwas defined as: vi=uipi. The following proposition
shows that at optimality, net utility follows the same order as intrinsic utility.
Proposition 4. Let pbe the price of an optimal solution of the Joint Assortment and Pricing
Problem. The following condition holds: uip
iui+1 p
i+1 i[γ(p)1].
The above propositions make it possible to filter out non-efficient assortments and prices by
restricting the search space to intrinsic utility ordered assortments and providing insights on how
the optimal solution behaves regarding prices and their relation with utilities. Based on these
propositions, the joint assortment and pricing optimisation problem for the TLM can be written
in a more succinct way. From Proposition 2, the solution is an intrinsic utility ordered set Sk= [k]
for some kn. Suppose there exists an optimal solution in the form (Sk, p) for a fixed value k.
In that case, recall that it is sufficient to restrict to valid pairs (Sk, p), meaning that c(Sk, p) = Sk.
Consider a fixed kn. By Proposition 4, at optimality, uipiujpj1i < j k.
Therefore, the condition that c(Sk, p) = Skcan be written as
gij (p) := exp(uipi)(1 + t)·exp(ujpj)0,1i < j k(14)
As a result, the joint k-assortment and pricing optimisation problem for the TLM (JAPTLM-k),
which aims at finding an optimal assortment Skof size kwith kn, can be written as:
maximize
pR(k)(p) := PiSkpi·exp(uipi)
PiSkexp(uipi) + a0
subject to gij (p)0,1i < j k
(JAPTLM-k)
Note that, if exp(u1uk)(1 + t), then the solution is the same as the unconstrained case,
because any fixed price can be assigned without creating dominances. Hence, the optimal revenue
R(k)can be calculated using equation (12), and all prices are equal to 1 + R(k). On the other
hand, if exp(u1uk)>1 + t, as in Example 6, the prices need to be adjusted in order to avoid
dominances.
The next theorem is the main result of this section.
Theorem 7. Problem JAPTLM-k can be solved in polynomial time.
The intuition behind the proof is based on Proposition 4 and the study of the Lagrangean
relaxation of problem (JAPTLM-k). Observe that, since uipiujpj(ij) at optimality,
23
then the largest ratio between attractiveness is obtained for products 1 and k. This ratio can also
occur for more products but only if they have the same net utility as products 1 or k. Thus, it must
be the case that there are non-negative integers k1and k2with k1+k2k, such that letting I1= [k1]
and I2={kk2+ 1, k k2+ 2, . . . , k}, the set of constraints C(k1, k2) = {gij(p)|iI1, j I2}
are satisfied at equality for the optimal solution (see the proof in Appendix A for details). Since it
is only necessary to study a polynomial number of combinations of constraints satisfied at equality
and, for each one of those combinations a closed form solution is provided, the result follows.
For the non-trivial case with exp(u1uk)>1 + t, where a fixed price fails to be optimal, the
prices need to be adjusted in order to avoid the dominances. Let R(k)and p(k)be the optimal
revenue and price vector. The following Lemma characterizes the structure of the optimal solution
for problem JAPTLM-k.
Lemma 3. The optimal solution to problem (JAPTLM-k)is either the same as the unconstrained
case (i.e. fixed price, in the case that exp(u1uk)(1 + t)) or the following holds at optimality:
a1(p1)
ak(pk)= 1 + t. (15)
Moreover, there are non-negative integers k
1, k
2, with k
1+k
2ksuch that:
R(k)=W
k
1+k
2
1+t·exp (1+t)PiI1ui+PiI2ui+k
2ln(1+t)
k
1(1+t)+k
21+Pi¯
Ikexp(ui1)
a0
,
where I1= [k
1],I2={kk
2+ 1, k k
2+ 2, . . . , k}and ¯
Ik= [k]\(I1I2). The optimal prices can
be obtained as follows:
p(k)
i=
1 + R(k)+ui(1+t)PiI1ui+PiI2ui+k
2ln(1+t)
k
1(1+t)+k
2if iI1,
1 + R(k)+ui(1+t)PiI1ui+PiI2ui+k
2ln(1+t)
k
1(1+t)+k
2+ ln(1 + t)if iI2,
1 + R(k)if i¯
Ik.
(16)
Let TLM-Opt(X, u, a0, k)be the procedure to obtain the optimal solution for problem (JAPTLM-k).
Using TLM-Opt(X, u, a0, k)at most ntimes (once for each kn) to obtain the assortment and
prices yielding the highest R(k), one can find the optimal assortment and price vector for any given
instance. Its intuition is to mimic the optimal strategy for the regular MNL (Fixed-Price Policy)
as much as possible. However, given that it needs to accommodate prices in order to avoid domi-
nances, the algorithm adjusts prices for the higher intrinsic utility products (making prices larger,
hence less attractive) and reduces the price of lower intrinsic utility ones, making them more attrac-
tive for customers and preventing them from being dominated. This allows the optimal strategy
to have an edge over strategies ignoring the Threshold induced dominances, such as Fixed-Price
Policy and, to a lesser extent, the Quasi-Same Price (Wang and Sahin, 2018). The Quasi-Same
Price policy policy only adjusts the price of the lowest attractiveness product, instead of adjusting
24
both extremes of the attractiveness spectrum and potentially multiple products.
8 Conclusion and Future Work
This paper studies the assortment optimization problem under the Two-Stage Luce model (2SLM),
a discrete choice model introduced by Echenique and Saito (2018) that generalizes the standard
multinomial logit model (MNL) with a dominance relation and may violate regularity. The paper
proved that the assortment problem under the 2SLM can be solved in polynomial time. The paper
also considered the capacitated assortment problem under the 2SLM and proved that the problem
becomes NP-hard in this setting. We also provide polynomial-time algorithms for special cases
of the capacitated problem when (1) the dominance relation is utility-correlated and when (2)
its transitive reduction is a forest. We also provide an Appendix showing numerical experiments
to highlight the performance of the proposed algorithms against classical strategies used in the
literature.
There are at least five interesting avenues for future research. First, one may wish to study how
to generalize the 2SLM further while still keeping the assortment problem solvable in polynomial
time. For example, one can try to check whether there exists a model that unifies the 2SLM and
the elegant work in Davis, Gallego, and Topaloglu (2013) where the assortment problem is still
solvable in polynomial time. Second, given that the capacitated version of the 2SLM is NP-hard
under Turing reductions (Theorem 4), it is interesting to see whether there exist good approximation
algorithms for this problem. Third, one can explore different forms of dominance. For example, one
may consider dominances specified by a discrete relation or a continuous functional form between
products. Fourth, one can try to generalise our results for the Joint Assortment Pricing Problem
under the Threshold Luce model to a more general setting, where price sensitivities depend on each
product. Finally, one can try to mix attention models with dominance relations, meaning that a
customer first perceives a subset of the products, dictated by an attention filter, and then filter the
products even more using dominance relations.
9 Acknowledgements
We thanks Yuval Filmus for his helpful insights leading us to find useful literature on this topic.
Thanks are also due to Guillermo Gallego for suggesting extending our assortment results to the
GAM model, and to Flavia Bonomo for relevant discussions..
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30
A Proofs
In this section we provide the proofs missing from the main text.
Proof of Theorem 4. The proof considers four problems:
1. Problem (MWBISBP): Maximum weighted independent set of size at most Cfor bipartite
graphs.
2. Problem (MWEBISBP): Maximum weighted independent set of size equal to Cfor bipartite
graphs.
3. Problem (EC2SLMAP): Optimal assortment under the General Luce model of size C.
4. Problem (C2SLMAP): Optimal capacitated assortment under the Two-Stage Luce model of size
at most C.
The proof shows that Problems (MWEBISBP), (EC2SLMAP), and (C2SLMAP) are NP-hard, using the
NP-hardness of Problem (MWBISBP) (Kalra et al., 2017) as a starting point.
First observe that Problem (MWEBISBP) is NP-hard under Turing reductions. Indeed, Problem
(MWBISBP) can be reduced to solving Cinstances of Problem (MWEBISBP) with budget c(1 cC).
We now show that Problem (EC2SLMAP) is NP-hard. Consider Problem (MWEBISBP) over a
bipartite graph G= (V=V1V2, E), where V1V2=, every edge (v1, v2)Esatisfies
v1V1and v2V2,wvis the weight of vertex v, and Cis the budget. We show that Problem
(MWEBISBP) over this bipartite graph can be polynomially reduced to Problem (EC2SLMAP). The
reduction assigns each vertex vto a product with a(v) = 1 and rv=wv, sets a0= 0, and has a
capacity C. Moreover, the reduction uses the following dominance relation: v1v2iff (v1, v2)E.
This dominance relation is irreflexive, anti-symmetric, and transitive, since the graph is bipartite. A
solution to Problem (MWEBISBP) is a feasible solution to Problem (EC2SLMAP), since the independent
set cannot contain two vertices v1, v2with v1v2by construction. Similarly, a feasible assortment
is an independent set, since the assortment cannot select two vertices v1V1and v2V2with
(v1, v2)E, since v1v2. The objective function of Problem (EC2SLMAP) reduces to maximizing
1
CX
vV
rvxv
which is equivalent to maximizing PvVrvxvsince exactly Cproducts will be selected by every
feasible assortment. The result follows by the NP-hardness of Problem (MWEBISBP).
Finally, Problem (C2SLMAP) is NP-hard under Turing reductions. Indeed, Problem (C2SLMAP)
can be reduced to solving Cinstances of Problem (EC2SLMAP) with capacity c(1 cC).
Proof of Theorem 5 By Theorem 2, it suffices to show that Problem (C-MaxAtt) is solved by
the recurrences in polynomial time. The correctness of recurrence A(v , c) comes from the fact that
vertex vdominates all its descendants and cannot be present in any assortment featuring any of
them. The correctness of recurrence A+(S, c) follows from the fact that eis not dominated by, and
i
does not dominate, any element in S, since they are all children of the same node. This also holds
for the descendants of eand the descendants of the elements in S. Hence, the optimal assortment is
obtained by splitting the capacity cinto n1and n2and merging the best assortment for A+(S, n1)
and A(e, n2) for some n1, n20 summing to c. The recurrences can be solved in polynomial time
since the computation for each vertex vand capacity ctakes O(n C) time, giving an overall time
complexity of O(n2C2).
Proof of Proposition 1. We prove this by contradiction. Suppose p
i< Rfor some iS, then
ˆ
S=S\ {i}has better revenue than the optimal solution if we keep the same prices and p
i< R.
Indeed, let us calculate R(ˆ
S):
R(ˆ
S) = Pjˆ
Seujp
j·p
j
Pjˆ
Seujp
j+a0
R(ˆ
S) =PjSeujp
j·p
jeuip
i·p
i
PjSeujp
jeuip
i+a0
R(ˆ
S) = PjSeujp
j·p
j
PjSeujp
j+a0
·PjSeujp
j+a0
PjSeujp
jeuip
i+a0
euip
i·p
i
PjSeujp
jeuip
i+a0
R(ˆ
S) = PjSeujp
j·p
j
PjSeujp
j+a0
·"1 + euip
i
PjSeujp
jeuip
i+a0#euip
i·p
i
PjSeujp
jeuip
i+a0
R(ˆ
S) =R·"1 + euip
i
PjSeujp
jeuip
i+a0#euip
i·p
i
PjSeujp
jeuip
i+a0
R(ˆ
S) =R+euip
i
PjSeujp
jeuip
i+a0
·[Rp
i]
| {z }
Γ
Now Γ is positive because p
i< R, but this implies R(ˆ
S)> R, contradicting the optimality
of R.
Proof of Proposition 2. Let (S, p) be an optimal solution. We can assume that (S, p)∈ V.
We proceed by contradiction. Suppose that there is a product inot included in the optimal solution
and another product jwith smaller intrinsic utility included in S. We show that we can include
product i, and remove jand get a greater revenue. Let ˆ
S= (S\ {j})∪ {i}, be the set where
we removed product j, and included product i. Let ˆpi=uiuj+p
j, this means that the total
attractiveness remains unchanged, and no new domination relations appear, given that product j
already had the same level attractiveness that product inow has. Observe that given that uiuj,
we have that ˆpip
j. Let us calculate R(ˆ
S, ˆp), where ˆpis the same as p, but with the proposed
changes in price:
ii
R(ˆ
S, ˆp) = Pkˆ
Seukˆpk·ˆpk
Pkˆ
Seukˆpk+a0
R(ˆ
S, ˆp) = PkSeukp
k·p
keujp
j·p
j+euiˆpi·ˆpi
Pkˆ
Seukˆpk+a0
R(ˆ
S, ˆp) = PkSeukp
k·p
k
PkSeukˆpk+a0
| {z }
R
+euiˆpi·ˆpieujp
j·p
j
Pkˆ
Seukˆpk+a0
R(ˆ
S, ˆp) = R+eujp
j
Pkˆ
Seukˆpk+a0
| {z }
0
·ˆpip
j
| {z }
>0
R(ˆ
S, ˆp)> R
Where we first rewrite R(ˆ
S, ˆp) using (S, p) because we just swapped product ifor product
j, and the total attractiveness remain the same, so the denominator does not change. Then we
identify R(S, p), and we use uiˆpi=ujpjto being able to factorize the remaining terms. So
we found a pair ( ˆ
S, ˆp), yielding strictly more revenue than (S, p), but adding product i, which
contradicts the optimality of (S, p).
Proof of Lemma 2. The proof (due to Wang and Sahin (2018)) is useful because it provides
intuition on how the optimal price variates when constrained to a fixed additive market share
among any two products. By the equality constraint, we have pj=ujln(Texp(uipi)), so
H(pi, pj) can be rewritten purely as a function of pias:
H(pi) = pi·exp(uipi)+(ujln(Texp(uipi))) ·(Texp(uipi)).(17)
Now, let us calculate the first derivative of H(pi) w.r.t. pi:
∂H (pi)
∂pi
= (pi+ (ujln(Texp(uipi)))) ·exp(uipi) (18)
Clearly the left-hand side term on the multiplication is monotonically decreasing from positive to
negative values as piincreases from 0 to . Therefore H(pi) is strictly unimodal and reaches its
maximum value at:
p
i=p
j= ln ((exp(ui) + exp(uj))/T ).
Proof of Proposition 3. We prove this result by contradiction. Let ibe the first index where
this condition does not hold, this means that p
i< p
i+1. Using Lemma 2, we found ˆpsatisfying
p
i<ˆp < p
i+1. Does this new price alter the consideration set? We show that this is not the case.
Indeed, the effect is two-fold: the price for product iincreases, and the price for product i+ 1
iii
decreases. We analyse the effect of these two consequences:
Increase on price for product i: This means a(i, p) decreases. Note that uiˆpui+1 p
i+1,
so neither ii+ 1 or i+ 1 i, because their attractiveness are now even closer than before.
Can ibe dominated now by another product? No, because given that uiui+1 we have
uiˆpui+1 ˆpui+1 p
i+1. Therefore the new attractiveness of iis still larger than the
new attractiveness of i+ 1, and the last inequality implies that the new attractiveness of iis
larger than the old attractiveness of i+ 1, and i+ 1 was not previously dominated either by
any other product.
Decrease on price for product i+ 1: Previously i+ 1 was not dominated by any product. Can
i+ 1 be dominated now? No, because if i+ 1 was not dominated before, now with a smaller
price ˆpits attractiveness is larger and therefore can’t be dominated now either (the only other
product that changed attractiveness was i, and it now has smaller attractiveness). Can i+ 1
dominate another product now with its new higher attractiveness? No, because given that
uiui+1 we have uip
iui+1 p
iui+1 ˆp, so the old attractiveness of product i
is larger than the new attractiveness of product i+ 1, and given that idid not dominate
another product before, the new price does not make i+ 1 dominate another product either.
So, letting pfix exactly the same as p, but replacing both p
iand p
i+1 with ˆp, means that the
pair (S, pfix) yields strictly more revenue than (S, p) (by Lemma 2), contradicting the optimality
assumption. The fact that equal intrinsic utility implies equal price at optimality, can be easily
demonstrated by the following: if two equal intrinsic utility products have different prices, then
using Lemma 2 we obtain strictly better revenue by assigning them the same price, and no new
domination occurs, because the new price is confined between the previous prices.
Proof of Proposition 4. We prove this by contradiction. Let pbe the optimal solution and i
be the first index where this condition does not hold. This means that uip
i< ui+1 p
i+1. We
can extrapolate this inequality further and say:
ui+1 p
i< uip
i< ui+1 p
i+1 < uip
i+1,(19)
because uiui+1 and pipi+1 by Propositions 2 and 3 respectively. We now do the following:
Define p0
iand p0
i+1 such as exp(uip0
i) + exp(ui+1 p0
i+1) = exp(uip
i) + exp(ui+1 p
i+1) and
exp(uip0
i) = exp(ui+1 p0
i+1). This means that:
p0
i=uiln exp(uip
i) + exp(ui+1 p
i+1)
2
p0
i+1 =ui+1 ln exp(uip
i) + exp(ui+1 p
i+1)
2
Consider H(pi, pi+1) = pi·exp(uipi) + pj·exp(ui+1 pi+1 ), where exp(uipi) + exp(ui
pi) = exp(uip
i) + exp(ui+1 p
i+1). By Lemma 2, H(pi, pi+1) is strictly increasing in pifor
iv
piˆpand strictly decreasing for piˆp, with ˆp= ln exp(ui)+exp(ui+1)
exp(uip
i)+exp(ui+1p
i+1)the solution of
the corresponding maximization problem of Lemma 2. We can verify that ˆp < p0
i< p
i. The first
inequality is straightforward. Indeed:
p0
i=uiln exp(uipi) + exp(ui+1 pi+1)
2
p0
i= ln 2 exp(ui)
exp(uip
i) + exp(ui+1 p
i+1)
p0
i>ln exp(ui) + exp(ui+1)
exp(uip
i) + exp(ui+1 p
i+1)
| {z }
ˆp
p0
i>ˆp
proving the desired inequality. Now, for the second one:
p0
i=uiln exp(uipi) + exp(ui+1 pi+1)
2
p0
i= ln 2 exp(ui)
exp(uip
i) + exp(ui+1 p
i+1)
p0
iln 2 exp(ui)
exp(uip
i) + exp(uip
i+1)
p0
i= ln 2 exp(ui)
exp(ui)(exp(p
i) + exp(p
i+1))
p0
i<ln 2
2 exp(p
i)
p0
i< p
i,
thus we have:
p0
i·exp(uip0
i) + p0
i+1 ·exp(ui+1 p0
i+1)> p
i·exp(uip
i) + p
i+1 ·exp(ui+1 p
i+1).
Meaning that we have the same assortment, but with prices p0
iand p0
i+1 generating strictly more
revenue than the optimal prices, which is a contradiction. The only thing that we have left to show
that with these new prices we are still on the same consideration set. It would be enough to show
that the new net utilities are bounded by previous values of net utilities. Indeed, we can verify
that p
i+1 p0
i+1 p0
ip
i, by simply using the definitions. We also know, by hypothesis that
uip0
i=ui+1 p0
i+1, then uip0
i=ui+1 p0
i+1 ui+1 p
i+1. So even when the price of product
idecreased, the new attractiveness is bounded above by a previously existing attractiveness, thus
not changing the consideration set. By the same reasoning, ui+1 p0
i+1 =uip0
iuip
i, meaning
v
that the new attractiveness is bounded below by a pre-existing one, so i+ 1 is not dominated with
this new prices either. So the consideration set stays the same, concluding the proof.
Proof of Theorem 7. We first write problem (JAPTLM-k) in minimization form to directly apply
the Karush-Khun-Tucker conditions (KKT)(Karush, 1939).
minimize
pR(k)(p)
subject to gij (p)0,1i<jk
(20)
The associated Lagrangean function is:
Lk(p, µ) = R(k)(p) + X
1i<jk
µij ·gij (p),(21)
where µij 0 are the associated Lagrange multipliers. Recall that if exp(u1uk)(1 + t), the
optimal revenue R(k)can be calculated using equation (12), and the solution corresponds to a fixed
price policy as for the regular multinomial logit.
On the other hand, if exp(u1uk)>(1 + t), any fixed price causes product kto be dominated
by product 1. Thus, to include product kin the assortment we need to adjust the prices. Let
p= (p1, . . . , pk) be the optimal price vector for problem (20). Observe that it can’t happen that
a1(p1)
ak(pk)<1+t, since by Proposition 4, it will also means that a1(p1)
a2(p2)<1+tand using Lemma 2 we can
find ˆpsuch that assigning ˆpto products 1 and 2 yields a larger revenue (and no dominance relation
appears, since the attractiveness of product 1 was reduced, and the attractiveness of product 2
increased, but is still less than the one of product 1), which contradicts optimality. Therefore, g1k
must be satisfied with equality, meaning a1(p1)
ak(pk)= 1 + t.
Furthermore, at optimality it holds uipiujpjij(by Proposition 4), and thus the
biggest ratio between attractiveness is observed for products 1 and k, and is exactly equal to 1 + t.
This ratio can be replicated for other pairs of products, but only if they share the same net utility
(and thus attractiveness) to the one of products 1 or k. Therefore, it must be the case that there are
integers k1and k2with k1+k2k, such that all products in I1= [k1] share the same attractiveness
(a1(p1)) and all products in I2={kk2+ 1, k k2+ 2, . . . , k}share the same attractiveness as
well (ak(pk)). This means that the set of constraints C(k1, k2) = {gij(p)|iI1, j I2}are all
satisfied with equality at optimality.
We now study the derivative of equation (21) with respect to each price pito obtain the KKT
conditions. We here assume that the first k1values share the same net utility value, meaning
us=u1p1=uipiiI1, and for the last k2products, we also have the same value of net
utility, that we call uf, this is: uf=ukpk=uipiiI2. Where these two quantities satisfy:
usuf= ln(1 + t),
Let us write the derivatives of the Lagrangean depending on where the index ibelongs. If iI1,
vi
then:
dLk
dpi
=exp(uipi)
PjSkexp(ujpj) + a0
·hpi1R(k)(p)iexp(uipi)·X
jI2
µij ,(22)
if iI2, we have:
dLk
dpi
=exp(uipi)
PjSkexp(ujpj) + a0
·hpi1R(k)(p)i+ (1 + t) exp(uipi)·X
jI1
µji ,(23)
And finally, if i¯
Ik= [k]\(i1I2), the derivative takes the following form:
dLk
dpi
=exp(uipi)
PjSkexp(ujpj) + a0
·hpi1R(k)(p)i(24)
Observe that i¯
Ik,dLk
dpi= 0 =pi= 1 + R(k)(p), and the right hand side is not dependent
on i, so all products in ¯
Ikshare the same price, which we denote ¯p. We can rewrite all prices and
the revenue depending on usand ¯p, using the following relations:
1. iI1u1p1=uipi=pi=uius
2. iI2u1p1=uipi+ ln(1 + t) =pi=uius+ ln(1 + t)
Note now that at optimality, for a fixed k, prices are determined by k1and k2. Thus, the
optimal revenue can be written explicitly depending on k,k1and k2, taking the following form:
R(k)(k1, k2) =
PiI1(uius) exp(us) + ¯pexp(¯p)Pi¯
Ikexp(ui) + PiI2(uius+ ln(1 + t)) exp(usln(1 + t))
PiI1exp(us) + exp(¯p)Pi¯
Ikexp(ui) + PiI2exp(us+ ln(1 + t)) + a0
(25)
Note that ¯p= 1 + R(k)(k1, k2) (Equation (24)) and let E(k1, k2) = Pi¯
Ikexp(ui). Using these
two relations, we can rewrite the optimal revenue as:
R(k)(k1, k2) =
eusP
iI1
(uius) + eus
1+t·P
iI2
(uius+ ln(1 + t)) + E(k1, k2)(1 + R(k)(k1, k2))e(1+R(k)(k1,k2))
eushk1+k2
1+ti+E(k1, k2)e(1+R(k)(k1,k2)) +a0
(26)
Up to this point, we have an equation relating the optimal revenue R(k)(k1, k2) and us. From
equation (22), after reordering terms we have:
pi1R(k)(k1, k2)
eus(k1+k2(1 + t)) + E(k1, k2)e(1+R(k)(k1,k2)) +a0
=X
jI2
µij ,iI1
uius1R(k)(k1, k2)
eus(k1+k2(1 + t)) + E(k1, k2)e(1+R(k)(k1,k2)) +a0
=X
jI2
µij ,iI1(27)
vii
Analogously, from equation (23), after reordering terms we have iI2:
pi1R(k)(k1, k2)
eus(k1+k2(1 + t)) + E(k1, k2)e(1+R(k)(k1,k2)) +a0
=(1 + t)X
jI1
µji ,iI2
1
1 + t·uius+ ln(1 + t)1R(k)(k1, k2)
eus(k1+k2(1 + t)) + E(k1, k2)e(1+R(k)(k1,k2)) +a0
=X
jI1
µji ,iI2(28)
Now, if we add equations (27) iI1then take equations (28) and also add them iI2, and
add those two results we can derive the value R(k)(k1, k2) as follows.
X
iI1
jI2
µij X
iI1
jI2
µij
| {z }
0
=PiI1(uius1R(k)(k1, k2)) + PiI2(uius+ln(1+t)1R(k)(k1,k2))
1+t
eus(k1+k2(1 + t)) + E(k1, k2)e(1+R(k)(k1,k2)) +a0
R(k)(k1, k2)k1+k2
1 + t=X
iI1
ui+1
1 + t·X
iI2
ui(1 + us)·k1+k2
1 + t+k2ln(1 + t)
1 + t
R(k)(k1, k2) =(1 + t)PiI1ui+PiI2ui+k2ln(1 + t)
k1(1 + t) + k2
1us(29)
We now have two equations relating R(k)(k1, k2) and usin (26) and (29). Using these equations
we can find the values of the optimal revenues and all the pricing structure while varying k1and
k2. If we define the following constant:
C1(k1, k2) = (1 + t)PiI1ui+PiI2u